I've also had a pairwise comparison calculation going on the web for a
while at http://www.slack.net/~whelan/cgi-bin/tbrw.cgi?rankings$pwc
(you'll have to scroll up to get the real PWC table if your browser
doesn't understand linking by ID). Mine currently differs from
USCHO's as well as Charlie's in that I'm not including Air Force as a
TUC pending the CHA tournament (the logic behind this is that they
have the same status as a team with a losing record which can become a
TUC by winning their conference tournament; their status as a TUC
depends on the outcome of the postseason).
Incidentally, the page contains the following:
1. A comparison of rankings based on the NCAA's criteria with
rankings based on the Bradley-Terry model (which self-consistently
gauges the strength of a team's opponents). This ranks the 54 D1
teams in order of KRACH and includes:
a) KRACH and rank (and implied Round-Robin Winning Percentage or
"RRWP")
b) RPI and rank
c) Overall record and winning percentage
d) PWR and rank (with Air Force not included as a TUC for the
time being)
e) "KPWR" and rank (this is a modification of PWR which uses
Bradley-Terry methods to correct for strength of schedule as
described below)
2. A table of all the pairwise comparisons using the NCAA's method.
Clicking on any of these PWCs will bring up a pop-up window listing
the outcome of each criterion, for instance, to see how Niagara beats
BU, click on the "BU" in Niagara's row and you'll get a popup
containing
Niagara Boston Univ
11-5-1.567 RPI .583 14-6-5
3-3 .500 TUC .577 5-3-5
11-4-1.719 L16 .656 8-3-5
5-2 .714 COp .667 4-2
1-0 H2H 0-1
Showing that BU has a better record against TUC and a better RPI, but
Niagara has a better record in the past 16 games, and one head-to-head
win, which decides the comparison. (This is easier to see with the
color-coding in the web version, BTW.)
3. A table of all the KPWCs, or Bradley-Terry modified pairwise
comparisons. The modification replaces RPI with KRACH (or
equivalently RRWP), and also modifies the "vs TUC", "Last 16" and
"Common Opponents" criteria to account for the strength of the
opponents each team played. (The "strength" of an opponent is
measured using their KRACH.) Again, clicking on a comparison gives a
popup with a rundown of all the criteria, for example, clicking on "Ni
in the "Boston Univ" row gives
Boston Univ Niagara
14-6-5 368.4KRACH 169.7 11-5-1
5-3-5 436.5 TUC 150.1 3-3
8-3-5 478.6 L16 202.1 11-4-1
4-2 206.5 COp 178.6 5-2
0-1 H2H 1-0
BU takes every criterion except for head-to-head, in the modified
system. If you'd like to see where those numbers come from, clicking
on the team's name at the front of a row gives a table containing the
teams' opponents, their KRACH ratings, and what results against each
team apply to the "vs TUC" and "Last 16" criteria. For instance,
here's Niagara's table:
Opponent KRACH Overall TUC Last 16
Boston Univ 368.4 1-0 1-0 1-0
St Lawrence 269.6 0-1 0-1 0-1
Colgate 243.6 0-1 0-1 0-1
AK-Anchorage 189.5 0-1 0-1
RPI 185.2 1-0 1-0 1-0
Princeton 121.1 1-0 1-0 1-0
Merrimack 102.4 1-0 1-0
Cornell 96.85 1-0 1-0
Vermont 90.59 0-1 0-1
Brown 42.11 1-0-1 1-0-1
Union 41.36 0-1 0-1
Quinnipiac 33.44 1-0 1-0 1-0
Air Force 18.05 3-0 3-0
Holy Cross 2.161 1-0 1-0
Criterion Rating 169.7 235.8 202.1
Won-Lost-Tied 11-5-1 4-3 11-4-1
(Note that the TUC rating here is not the same as that used when
making a comparison with BU, since the latter has the head-to-head
BU/Niagara game removed.) To explain where these "criterion ratings"
come from, look at Niagara's rating for the TUC criterion. They
accumulated a 4-3 record playing one game each against teams with
KRACHs of 368.4, 269.5, 243.6 189.5, 185.2, 121.1 and 33.44. A team
with a KRACH of x is expected to collect x/(x+y) of the points in
games against a team with a KRACH of y, so Niagara's TUC rating of
235.8 is chosen to predict that they would win 4 out of 7 games played
against those 7 teams. That is,
235.8 235.8 235.8 235.8
----------- + ----------- + ... + ----------- + ----------- = 4
235.8+368.4 235.8+269.5 235.8+121.1 235.8+33.44
To show why this is a good idea, look at the common opponents
criterion for BU and Niagara (this table is not on the web; I had to
examine BU and Niagara's pages side-by-side to produce it).
Opponent KRACH Niagara Boston Univ
AK-Anchorage 189.5 0-1 1-0
RPI 185.2 1-0 0-1
Merrimack 102.4 1-0 2-0
Vermont 90.59 0-1 0-1
Air Force 18.05 3-0 1-0
Criterion Rating 178.6 206.5
Won-Lost-Tied 5-2 4-2
So we see how, even though this is a common opponents criterion, with
BU and Niagara by definition playing the same opponents, their
strength of schedule for the criterion is different. If each team had
played each of their common opponents only once, they would both have
3-2 records and identical Criterion Ratings. The difference between
them is that Niagara had two additional wins over Air Force and BU had
one additional win over Merrimack. Since Merrimack is about half as
good as either BU or Niagara, while Air Force is only 1/10 as good
(according to KRACH), one win over Merrimack is more impressive than
two over Air Force, and thus BU has a higher criterion rating although
Niagara has a higher winnin percentage in those games.
The adjustments for strength of schedule are obviously much more
substantial for the Last 16 and vs TUC criteria, where the strength of
the teams' opposition can vary a lot.
The current top 12 in the two schemes is as follows:
Plain PWR Modified=KPWR
1. New Hampshire New Hampshire
2. Wisconsin Wisconsin
3. Maine North Dakota
4. Michigan Maine
5. North Dakota Michigan
6. Niagara Boston University
7. Boston University Minnesota
8. Quinnipiac Northeastern
9. Northeastern St. Cloud
10. Michigan State Boston College
11. Ferris State Michigan State
12. Colgate Colgate
The scores for this analysis and that appearing on the web include
game results through January 24 and come from USCHO's Division I
composite schedule,
http://www.uscollegehockey.com/cgi/confsched.cgi?season=19992000&gender=m&divisi
on=I
John Whelan, Cornell '91
[log in to unmask]
http://www.amurgsval.org/joe/
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